Nondeterminism makes unary 1-limited automata concise (2504.08464v2)

Abstract: We investigate the descriptional complexity of different variants of 1-limited automata (1-las), an extension of two-way finite automata (2nfas) characterizing regular languages. In particular, we consider 2nfas with common-guess (2nfas+cg), which are 2nfas equipped with a new kind of nondeterminism that allows the device to initially annotate each input symbol, before performing a read-only computation over the resulting annotated word. Their deterministic counterparts, namely two-way deterministic finite automata with common-guess (2dfas+cg), still have a nondeterministic annotation phase and can be considered as a restriction of 1-las. We prove exponential lower bounds for the simulations of 2dfas+cg (and thus of 1-las) by deterministic 1-las and by 2nfas. These results are derived from a doubly exponential lower bound for the simulation of 2dfas+cg by one-way deterministic finite automata (1dfas). Our lower bounds are witnessed by unary languages, namely languages defined over a singleton alphabet. As a consequence, we close a question left open in [Pighizzini and Prigioniero. Limited automata and unary languages. Inf. Comput., 266:60-74], about the existence of a double exponential gap between 1-las and 1dfas in the unary case. Lastly, we prove an exponential lower bound for complementing unary 2dfas+cg (and thus unary 1-las).